314 XV NORMED ALGEBRAS AND SPECTRAL THEORY (a) Let A be a Banach algebra, and put b = inf (\\x2 \\l\\x\\2). Show that **0 X for all x E A. (b) If x, y are two elements of a Banach algebra A, show that the intersections of Sp(xy) and $p(yx) with C — {0} are equal, and hence that p(xy) = p(yx) (cf. Section 11.1, Problem 2). 2. Let A be a Banach algebra with unit element. (a) If x e A has a left (resp. right) inverse y, then each x' e A such that I* - x\\ <\\y\\-1 has a left (resp. right) inverse. (b) Let (xn) be a sequence of elements of A, each of which has a left (resp. right) inverse yn . If the sequence (#„) converges to x, and if the sequence (yn) is bounded, then x has a left (resp. right) inverse. 3. Let A be a normed algebra. For each x e A, let Lx (resp. Rx) denote the continuous linear mapping y \-+xy (resp. y\-*yx) of A into A. The element x is said to be a left (resp. right) topological zero-divisor if Lx (resp. Rx) is not a homeomorphism of A onto its image. (a) Give an example of a left topological zero-divisor which is not a left zero-divisor (cf. Section 11.1, Problem 4). (b) For each x e A, put A(x) = inf ||xy||/|M|, A'(x) = inf \\yx\\l\\y\\. y#Q y?fcO Show that |A(*) - AW! g || x~ yl |A'(jc) - A'GOI < |jjc - y\\, X(x)\(y) Deduce that the set of left (resp. right) topological zero-divisors is closed in A. (c) Let A be a Banach algebra with unit element. If x e A is not left-invertible but is the limit of a sequence (xn) of left-invertible elements, then x is a right topological zero divisor (use Problem 2(b)). Deduce that the set of elements of A which are neither invertible nor topological zero-divisors is open in A. (d) In the algebra jtf(X) (15.15), show that the identity function lx is neither invertible nor a topologicai zero-divisor. 4. Let A be a Banach algebra with unit element e, in which the only right topological zero-divisor is 0. Show that A == Ce. (Consider the frontier points of Sp(x) and use Problem 3(c).) 5. (a) In a Banach algebra A with unit element, an element x is said to be topologically nilpotent if the sequence (xn)n^1 tends to 0. Show that x is topologically nilpotent if and only if p(x) < 1. (b) An element x e A is said to be quasi-nilpotent if p(x) = 0, or equivalently if (A*)" tends to 0 as n tends to 4- <x>, for all scalars A. A quasi-nilpotent element is a left and right topological zero-divisor. (c) Let u be a continuous endomorphism of a Banach space E. If lim ||w"(f)|| 1/fl = 0 rt-»00 for all / e E, show that u is quasi-nilpotent in the Banach algebra A = -S?(E). (Observeugh positive values, >(FM(x0 4- /») tends to a limit equal to 7rg(x0).d. Then, for each neighborhood U of e, there exists a number e > 0 such that